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\def\Tr{\mathop{\rm Tr}\nolimits}
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\def\ketx{|(x_i)\rangle}
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\def\gxex{\bragx\ebh\ketx}
\def\sumpi{\sum_{[p]}}
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\begin{document}
\title{Percolation transition in the Bose gas}
\author{Andr\'as S\"ut\H{o}\thanks{Permanent address: Central Research Institute
for Physics, Budapest}\\Institut de Physique Th\'eorique\\
Ecole Polytechnique F\'ed\'erale de Lausanne\\
CH-1015 Lausanne, Switzerland}
\date{ }
\maketitle
\begin{abstract}
The canonical partition function of a Bose gas gives rise to a probability
distribution over the permutations of $N$ particles. We study the probability
and mean value of the cycle lengths in the cyclic permutations, their relation
to physical quantities like pair correlations, and their thermodynamic limit.
We show that in the ground state of most interacting
boson gas the mean cycle length diverges in the bulk limit and the particles
form macroscopic cycles. In the free Bose gas Bose-Einstein condensation is
accompanied by a percolation transition: the appearance of infinite cycles with
nonvanishing probability.
\end{abstract}
\vspace*{2cm}
\hspace*{1.5cm}To appear in J. Phys. A
\newpage
\section{Introduction}
This paper presents a new approach to phase transitions in bosonic systems.
Since this description emerges somewhat accidentally from a
study of the ferromagnetism in the Hubbard model, it may be interesting
to outline the sequence of ideas connecting these seemingly distant fields.
The magnetization per particle in the Hubbard model (in fact, in any model of
spin-$1/2$ fermions) can be written ([AL], [S1], [S2]) as
\be\label{1.1}
M(h)=\frac{1}{2}\frac{\sum_{[p]}(\frac{1}{N}\sum_{j}p_j\tanh\frac{1}{2}
p_j\beta h)\epsilon[p]A[p]\prod_j(2\cosh\frac{1}{2}
p_j\beta h)}
{\sum_{[p]}\epsilon[p]A[p]\prod_j(2\cosh\frac{1}{2}
p_j\beta h)}\;\;.
\ee
Equation (\ref{1.1}) and the formulas (\ref{1.4}) and (\ref{1.6}) below will
be derived in full generality in Section 3. At this place let us concentrate
on the structure of the above expression. The summations run
over the partitions $[p]$ of $N$, the number of particles:
\be\label{1.2}
p_1\geq p_2\geq\ldots\geq 1\;\;,\;\;p_1+p_2+\cdots =N\;.
\ee
$\epsilon[p]=(-1)^{\sum(p_j-1)}$, $h$ is the external field, $\beta$ is the
inverse temperature and $A[p]$ (see Eq.(\ref{2.14})) is independent of $h$.
Each partition corresponds to a conjugacy class of the group $S_N$ of
permutations of $N$ elements and fixes the cycle lengths of the permutations
within the class [LF]. One observes that the magnetization is a rational
function of $\tanh\frac{1}{2}\beta h$, which can be expanded to give
\be
M(h)=\sum_{n=0}^{\infty}a_n(\tanh\frac{1}{2}\beta h)^{2n+1}\;.
\ee
The zero field magnetic susceptibility $\chi$ reads
\bea\label{1.4}
\beta^{-1}\chi=\frac{a_0}{2}=\frac{1}{4}\frac
{\sum_{[p]}(\frac{1}{N}\sum_jp_j^2)\epsilon[p]2^{k[p]}A[p]}
{\sum_{[p]}\epsilon[p]2^{k[p]}A[p]}
=\frac{1}{4}\langle\langle p_j\rangle\rangle\;,
\eea
where $k[p]$ is the number of elements of the partition $[p]$. The above
is a sort of mean value of the cycle lengths: The first average is taken with
the weights $p_j/N$, the second is the thermodynamic average. Let ${\bf S}(x)$
denote the spin operator at the site $x$, $S^z(x)$ its $z$-component and
$\bf S$ the operator of the total spin. Then the fluctuation-dissipation formula
\bea\label{1.5}
\beta^{-1}\chi=\frac{1}{N}\sum_{x,y}\langle S^z(x)S^z(y)\rangle =
\frac{1}{3N}\sum_{x,y}\langle{\bf S}(x){\bf S}(y)\rangle
=\frac{1}{3N}\langle{\bf S}^2\rangle
\eea
compared with (\ref{1.4}) shows that there is a ferromagnetic long-range order
if and only if the average cycle length grows like the number
of electrons when the thermodynamic limit is taken.
Spin correlations in zero field are connected to the cyclic permutations also
locally: For $x\neq y$,
\be\label{1.6}
\langle{\bf S}(x){\bf S}(y)\rangle=\frac{3}{4}\frac
{\sum_{g\neq e}\epsilon(g)2^{k(g)}\sum_{(x_1,\ldots ,x_N)}'
\langle x_{g(1)},... ,x_{g(N)}|e^{-\beta H_N}|x_1,... ,x_N\rangle}
{\sum_g\epsilon(g)2^{k(g)}\sum_{(x_1,\ldots,x_N)}
\langle x_{g(1)},...,x_{g(N)}|e^{-\beta H_N}|x_1,...,x_N\rangle}\;
\ee
(cf. Eq.(\ref{3.23})).
In this formula $H_N$ is the $N$-particle Hamiltonian and
\[|x_1,\ldots,x_N\rangle=\otimes|x_i\rangle\;;\]
$g$ denotes a permutation of
$1,2,...,N$, $\epsilon(g)$ is the signature and $k(g)$ is the number of
disjoint cycles of $g$, and $e$ is the unit of $S_N$. The primed sum goes over
$(x_1,...,x_N)$ such that $x_i=x$ and $x_j=y$ for a single pair $(i,j)$,
where $i$ and $j$ are in the same cycle of $g$. The remarkable fact about this
formula is that the numerator is a part of the sum constituting the denominator.
Were not $\epsilon(g)$ there, we could interprete the ratio as the
probability of finding a configuration $(x_i)$ and a permutation $g$ such that
$x$ and $y$ are singly occupied and the two particles belong to the same cycle
of $g$. Long-range ferromagnetic order then would mean that spatially extended
cycles have nonvanishing probability. This interpretation becomes possible if
we drop $\epsilon(g)$. This makes (\ref{1.4}) turn into a true mean value and
(\ref{1.6}) into a true probability. The system is now a boson gas with a
two-valued internal degree of freedom. If we replace
the numbers $1/4$, $3/4$ and $2$ respectively by $s(s+1)/3$, $s(s+1)$ and
$2s+1$, we obtain the susceptibility and pair correlation of a spin-$s$
boson gas. As a last step, we divide the two equations by $s$ and then set
$s=0$. The right-hand side of the first is still the mean value of the cycle
lengths, that of the second is still the probability quoted above. The obvious
physical meaning of the left-hand sides has been lost, however they must refer
to some thermodynamic properties of the spinless boson gas.
One observes that $\langle\langle p_i\rangle\rangle$ takes on values between
$1$ and $N$. At high temperatures and/or low densities
$\langle\langle p_i\rangle\rangle$ must be of the order of $1$ as $N$ and $V$
increase. This is at least clear for the Bose gas with spin, where in the
opposite case we would get non- or slowly decaying magnetic correlations at
arbitrarily high temperatures or low densities. The boundedness of
$\langle\langle p_i\rangle\rangle$ implies that in the thermodynamic limit all
the cycles are finite with full probability. This may not be the case at low
temperatures and/or high densities. If a transition occurs it will be a
percolation transition in the sense that infinite cycles appear with positive
density. This has thermodynamic consequences through the divergence of the
response function $\langle\langle p_i\rangle\rangle$.
The aim of this paper is to make the above ideas mathematically more precise.
In Section 2 we define a probability distribution $P_{V,N}$ over the symmetric
group $S_N$ by using the canonical partition function of a system of $N$
bosons in a volume $V$. Associated with the cyclic permutations in which any
permutation $g$ can be decomposed, we introduce two sets of random variables:
$\nu_k(g)$, the number of cycles of length $k$ and $\xi_i(g)$, the length of the
cycle containing the number $i$. The mean value of the first and the probability
distribution of the second are related by a simple expression. Section 3,
supplemented by an Appendix, contains the derivation of the formulas
(\ref{1.1}),(\ref{1.4}) and (\ref{1.6}) in full generality for bosons and
fermions. In particular, for the magnetic Bose gas the expectation value
$\evn(\xi_i)$ proves to be the zero field susceptibility
while in the nonmagnetic
case it is related to some special kind of density-density correlation
functions. In Section 4 we sketch the problem of the thermodynamic
limit of the probability distribution introduced in Section 2, and give the
definition of what we call the cycle percolation. Sections 5 and 6 present
examples.
In Section 5 we study $P_{V,N}$ at zero temperature for the interacting Bose
gas. We show that if the overall ground state of the Hamiltonian in the spinless
Hilbert space (without Bose or Fermi statistics) is unique, the corresponding
Bose gas exhibits cycle percolation in the ground state: $\evn(\xi_i)$ grows
proportionally to $N$ and the probability of cycle percolation is $1$. This
clearly shows the interest in this quantity: being zero at high temperature
and $1$ in the ground state, it is a good candidate for an order parameter.
Notice, in contradistinction, that the non-vanishing of the off-diagonal
long-range order (ODLRO, [Ya]) in the ground state of an interacting
Bose gas is apparently not easier to establish than to show Bose condensation
at positive temperatures. As a rare example, recently Penrose [Pe] proved
ODLRO for the hard-core Bose gas on the complete graph, a model solved
earlier by T\'oth [To1].
The prerequisite to ground state cycle percolation, the uniqueness of the
overall ground state in finite volumes holds true rather generally, for
example, in two and higher dimensions for pair interactions which are bounded
from below everywhere and from above outside the origin or a hard core.
(For hard-core interactions the density must be smaller than the closed-packing
value.) Strictly speaking, this is proved only for Dirichlet boundary condition
on an arbitrary connected domain. Relevant results are due to Rutman
([KR], Theorem 6.3), Glimm and Jaffe ([GJ], $\S$~3.3), Faris [Fa], Faris and
Simon [FS] and Simon ([Si], Theorem 21.1). Uniqueness is probably also true
for periodic and Neumann boundary condition on rectangular domains where the
proof is immediate for noninteracting particles. Natural counterexamples for
the uniqueness of the ground state are provided by one-dimensional systems
with hard-core or other pair interactions which are repulsive and nonintegrable
at the origin. Such interactions cut the phase space into $N!$ disconnected
parts ($(N-1)!$ if the boundary condition is periodic) so that there are
$N!$ linearly independent ground states. In the corresponding Fermi or Bose
gas the exchange is completely prohibited: at any temperature including zero
each particle forms a $1$-cycle in itself, the fermion and boson partition
functions coincide. So there is no cycle percolation and, indeed, no Bose
condensation in the case of attractive walls ([BP], [Sm]), although these
latter give rise to condensation in the 1D free Bose gas [Ro]. Notice that the
uniqueness condition replaces the absence of long-range order which was argued
to be necessary for ODLRO in the ground state [PO].
In Section 6 we show that in the free Bose gas Bose-Einstein condensation
calls forth the percolation transition. In this case the
probabilities $P_{V,N}(\xi_i=k)$ can be obtained explicitly and their
asymptotic behaviour for different boundary conditions can be studied. This
and other details will be published separately [S3].
Let us finish this introduction with several remarks.
A purely quantum mechanical phase transition in the Bose gas, that is, a phase
transition which is entirely due to Bose statistics
is driven by the exchange interaction.
Exchange acts among particles
which are cyclicly permuted. If $\evn(\xi_i)$ remains finite, the
symmetrisation plays
a minor role and Boltzmann statistics would give a qualitatively correct
description. It is only when $\evn(\xi_i)$ diverges
that Bose statistics becomes relevant.
Therefore, the divergence of $\evn(\xi_i)$ with increasing $N$ and $V$ is
probably the most general criterion of such a phase transition: more general
than ODLRO and even more general than cycle percolation. (Take, for instance,
$\pvn(\xi_i=n)=a_N/n^2$ with normalizing factor $a_N$, then
$\evn(\xi_i)\approx a_N\ln N$ diverges but
\[\sum_{n=1}^\infty\lim_{N\rightarrow\infty}\pvn(\xi_i=n)=
6/\pi^2\sum_{n=1}^\infty 1/n^2=1\]
and, hence, there is no cycle percolation, see Section 4.) On the other hand,
ODLRO implies cycle percolation in the free Bose gas and probably also
in interacting systems: The importance of long cycles in the
$\lambda$-transition in liquid helium was already observed by Feynman [Fe]
and Penrose and Onsager [PO]. If the mean cycle length diverges slowly
(more slowly than $N$), there may occur a phase transition analogous to the
Kosterlitz-Thouless transition, with or without ODLRO and cycle percolation.
Random walks in connection with path-integral representations of the partition
function are realisations of the cyclic permutations in "spacetime". It is in
these terms that Feynman [Fe] described the $\lambda$-transition in liquid
helium. Closely related ideas appear in recent works, mainly in
connection with the quantum Heisenberg model [CS], in particular in attempts
to describe the ground state of the two-dimensional antiferromagnet [Mi]
or the phase transition
in the three-dimensional spin-$1/2$ ferromagnet [To2]. In the set-up of the
present paper random walks have no conceptual importance, their introduction
can be avoided.
It is interesting to point out the role of the spins in the above description.
While spins are not thought to modify the nature of the phase transition,
their presence is useful, as they are the most natural markers of the cycles.
A preliminary version of this work was presented at the 18th IUPAP Conference
on Statistical Physics [S4].
{\bf Acknowledgement}. I would like to thank David Bridges, Herv\'e Kunz,
Philippe Martin, Andreas Mielke, Domokos Sz\'asz and B\'alint T\'oth for
discussions, comments and advises.
\section{Probability distribution over permutations}
\setcounter{equation}{0}
The canonical partition function of a system of $N$ bosons confined in a volume
$V$ can be written as
\be\label{2.1}
Q_{V,N}=\Tr P_+ e^{-\beta H_N}\;.
\ee
With some abuse of notation, $V$ will be used to denote both the domain and the
volume (the set of sites and their number in the lattice case). In
Eq.(\ref{2.1}) (and, unless otherwise stated, in all subsequent formulas)
the trace is taken in ${\cal H}^{\otimes N}$, the $N$-times tensor product
of the one-particle Hilbert space $\cal H$. The $N$-particle Hamiltonian is
\be\label{2.2}
H_N=-\frac{\hbar^2}{2m}\sum_{i=1}^N\Delta_i+u_N(x_1,\ldots,x_N)
\ee
where $m$ is the particle mass and $\Delta_i$ is the Laplacian acting in the
coordinates $x_i=(x_i^1,\ldots,x_i^D)$ of the $i$th particle. The potential
energy $u_N$ is a real symmetric function of its arguments.
The only assumption about $u_N$ is that it permits to define the trace of
$e^{-\beta H_N}$.
In Eq.(\ref{2.1}),
\be\label{2.3}
P_+=N!^{-1}\sum_{g\in S_N}U(g)\;.
\ee
$U$ is the unitary representation of the permutation group $S_N$ in
${\cal H}^{\otimes N}$; the action of $U(g)$ is defined by
\be\label{2.4}
U(g)|\psi_1,\ldots,\psi_N\rangle=|\psi_{g^{-1}(1)},\ldots,\psi_{g^{-1}(N)}
\rangle\;.
\ee
It is easy to verify that $P_+$ is self-adjoint and $P_+^2=P_+$, so that
$P_+$ is the orthogonal projection onto the symmetric subspace of
${\cal H}^{\otimes N}$. Substituting Eq.(\ref{2.3}) into Eq.({\ref{2.1}) we
obtain a sum over the permutation group and notice that the summand depends
only on the conjugation class to which $g$ belongs.
Indeed, let $g$ and $h$ be conjugate to each other, i.e., $h=fgf^{-1}$ for
some $f$ in $S_N$, then
\bea\label{2.5}
\Tr U(h)e^{-\beta H_N}=\Tr U(f)U(g)U(f)^{-1}e^{-\beta H_N}=\Tr U(g)
e^{-\beta H_N}\;.
\eea
The first equality holds because $U$ is a representation of $S_N$, the second
because of the cyclicity of the trace and because $U(f)$ commutes with $H_N$.
In virtue of Eq.(\ref{2.5}), we can rewrite $Q_{V,N}$ as
\be\label{2.6}
Q_{V,N}=N!^{-1}\sum_{\cal K}|{\cal K}|\Tr U(g)\ebh
\ee
where the summation runs over the conjugacy classes of $S_N$, $|\cal K|$ is the
number of elements in $\cal K$ and $g$ is any element of $\cal K$.
The class corresponding
to the partition $[p]$ (see (\ref{1.2})) consists of all the permutations
of the form
\be\label{2.7}
g=g_1g_2\ldots=(i_1\ldots i_{p_1})(i_{p_1+1}\ldots i_{p_1+p_2})\cdots
\ee
where $g_i$ are cyclic permutations of pairwise disjoint subsets
\be\label{2.7a}
C_1=\{i_1,\ldots,i_{p_1}\}\;,\;C_2=\{i_{p_1+1},\ldots,i_{p_1+p_2}\},\dots
\ee
of $\{1,\ldots,N\}$.
If $n_j\geq 0$ denotes the multiplicity of $j$ in $[p]$, the sequence
$(n_j)$ satisfies
\be\label{2.8}
\sum_{j=1}^Njn_j=N\;.
\ee
The relation between $[p]$ and $(n_j)$ is one-to-one, therefore the notation
\be\label{2.9}
q[p]=\prod_{j=1}^N\left(\frac{1}{j}\right)^{n_j}\frac{1}{n_j!}
\ee
is unambiguous. Now $N!q[p]$ is the number of elements of the class $[p]$,
thus Eq.(\ref{2.6}) reads
\be\label{2.30}
Q_{V,N}=\sum_{[p]}q[p]\Tr U(g)\ebh\;.
\ee
This formula was the starting point for Matsubara [Ma], Feynman [Fe] and
Penrose and Onsager [PO] in the discussion of the $\lambda$-transition in
liquid helium.
Let us remark that the canonical partition function of a system of fermions
has the form (\ref{2.30}) with an extra factor $\epsilon[p]$. This can be
obtained by replacing $P_+$ by
\be
P_-=N!^{-1}\sum_{g\in S_N}\epsilon(g)U(g)
\ee
in Eq.(\ref{2.1}) and by noticing that the signature $\epsilon(g)$ depends
only on the class: each $j$-cycle contributes to it with a factor $(-1)^{j-1}$.
In the simplest situation the one-particle Hilbert space is $L^2(V)$ (or
$\ell^2(V)$, lattice case). Another example is when the particles have an
internal degree of freedom (spin) which may take on $d=2s+1$ values. In this
case
\be\label{2.11}
{\cal H}={\cal H}_0\otimes{\Bbb C}^d
\ee
where ${\cal H}_0$ denotes the spinless one-particle Hilbert space. The
Hamiltonian acts exclusively in ${\cal H}_0$. Therefore the partial trace over
$({\Bbb C}^d)^{\otimes N}$ can be performed: Using the notation $U_0$ and
$U_1$ for the representations of $S_N$ in ${\cal H}_0^{\otimes N}$ and
$({\Bbb C}^d)^{\otimes N}$, respectively, we obtain
\bea\label{2.12}
\Tr U(g)e^{-\beta H_N}=\Tr_{({\Bbb C}^d)^{\otimes N}}U_1(g)
\Tr_{{\cal H}_0^{\otimes N}}U_0(g)e^{-\beta H_N}\;.
\eea
The first term on the right-hand side is the character $\chi_N(g)$ of $g$ in
the representation $U_1$. With the decomposition (\ref{2.7}), this factorises
according to the cycles,
\be\label{2.13}
\chi_N(g)=\prod_{i=1}^k\chi_{p_i}(g_i)=d^k\;,
\ee
see also Eq.(\ref{a.5}). Thus, for $\cal H$ given by Eq.(\ref{2.11}),
\be\label{2.14}
Q_{V,N}=\sum_{[p]}d^{k[p]}q[p]\Tr_{{\cal H}_0^{\otimes N}}U_0(g)\ebh
=\sum_{[p]}d^{k[p]}A[p]\;.
\ee
The number of cycles $k[p]=\sum n_j$ includes cycles of length $1$.
$A[p]$ is defined by Eq.(\ref{2.14}). It is this quantity which appeared
in Eq.(\ref{1.1}).
In what follows, we will consider $S_N$ as a space of events and assign
probabilities to the permutations. The probability of any $g\in S_N$ is defined
as
\be\label{2.15}
P_{V,N}(g)=(N!Q_{V,N})^{-1}\Tr U(g)\ebh\;.
\ee
To see that this expression is positive write
\be
\Tr U(g)\ebh=\int_V...\int_V dx_1...dx_N
\langle x_1,...,x_N|\ebh|x_{g(1)},...,x_{g(N)}\rangle\;.
\ee
The positivity of the integrand can be proved by showing it at first for the
free Hamiltonian $-\sum\Delta_i$ and then by passing to $H_N$ with the
application of the Trotter formula. For $-\sum\Delta_i$ the proof is done by
direct computation if the domain is rectangular and the boundary condition is
periodic or Neumann; for Dirichlet boundary condition on arbitrary domain
the proof involves path integral arguments ([Gi], [FS]).
To simplify the notation, the labels $V$ and $N$ will be dropped
whenever this causes no confusion. Next, we introduce two sets of random
variables: $\nu_j(g)$ is the number of $j$-cycles of $g$ and $\xi_i(g)$ is
the length of the cycle containing $i$ ($1\leq i,j\leq N$). These are related
by
\be\label{2.16}
\sum\xi_i=\sum j^2\nu_j\;.
\ee
To see this, observe that both sides depend only on the class $[p]$ and both
equal $\sum p_j^2$. All the $\xi_i$ are equally distributed, therefore
\be\label{2.17}
\xi=N^{-1}\sum\xi_i
\ee
has the same mean value as any of the $\xi_i$:
\be\label{2.18}
E_{V,N}(\xi)=E_{V,N}(\xi_i)=N^{-1}\sum j^2 E_{V,N}(\nu_j)\;.
\ee
Again, the labels $V$ and $N$ will often be dropped hereafter. $E(\xi)$ will be
referred to as the "mean cycle length", although the true mean cycle length
may be smaller:
\be\label{2.19}
\xi\geq N/\sum\nu_j\;.
\ee
Equation (\ref{2.19}) is equivalent with the Schwarz inequality
\be
N^2=(\sum j\nu_j)^2\leq (\sum\nu_j)(\sum j^2\nu_j)=N\xi\sum\nu_j\;.
\ee
There is a simple relation between the probability distribution of $\xi_i$
and the mean value of the $\nu_j$'s. It reads
\be\label{2.21}
P(\xi_i=n)=\frac{n}{N}E(\nu_n)\;.
\ee
Equation (\ref{2.18}) could have been obtained from here as well. To get
Eq.(\ref{2.21}) we write
\be
P(\xi_i=n)=\sum_j P(\xi_i=n|\nu_n=j)P(\nu_n=j)
\ee
and notice that
\be\label{2.23}
P(\xi_i=n|\nu_n=j)=\frac{jn}{N}\;.
\ee
A similar simple relation can be derived for the probability that any two
different numbers, say $i$ and $j$, fall in the same cycle of $g$. Let us denote
this event by $i\sim_g j$ or more simply by $i\sim j$.
\bea\label{2.24}
P(i\sim j)&=&\sum_n P(i\sim j|\xi_i=n)P(\xi_i=n)\nonumber\\
&=&\sum_n\frac{n-1}{N-1}P(\xi_i=n)=\frac{1}{N-1}(E(\xi)-1)\;.
\eea
To compute the conditional probabilities in Eqs.(\ref{2.23}) and (\ref{2.24})
we used the fact that $\pvn(g)$ is a class function (see Eq.(\ref{2.5})),
therefore a given number
can be found with equal probability in any box of a Young diagram.
If we replace the definition (\ref{2.15}) by any probability distribution
$P_N$ on $S_N$ which is constant on the conjugacy classes, the probability
of a class is
\be\label{2.31a}
P_N[p]=N!q[p]P_N(g)
\ee
where $g$ is any element of the class, all $\xi_i$ are equally distributed
and the equations (\ref{2.18}) and (\ref{2.21}-\ref{2.24}) remain valid.
Equation (\ref{2.21}) can be generalized as follows. Let $m_1,\ldots,m_k$
be $k$ different positive integers and $1\leq i_1<\cdotsFrom Eq.(\ref{1.5}) we get for any $i\neq j$
\bea\label{3.12}
\chi(s)=(\beta/3N)\langle\spin^2\rangle=\frac{1}{3}s(s+1)\beta
+\frac{1}{3}\beta(N-1)\sij
\eea
because $\sij$ is independent of $i,j$. Comparison with Eqs.(\ref{3.10}) and
(\ref{2.24}) shows that for $i\neq j$
\be\label{3.13}
\sij=s(s+1)P(i\sim j)\;.
\ee
Thus, the correlations in a gas of spinning bosons are ferromagnetic and
ferromagnetic long-range order means that the probability of finding $i$ and
$j$ in the same cycle is nonvanishing in the thermodynamic limit.
Equation (\ref{3.13}) contains no information about the spatial behaviour
of the pair correlations. This latter can be inferred from the analogue of
Eq.(\ref{1.6}). For the sake of simplicity, we restrict the discussion to the
lattice case. The spin operator at the site $x$ is defined by
\be
\spin(x)=\sum_{i=1}^N\spin_iN_i(x)\;.
\ee
$N_i(x)$ projects the position of particle $i$ to $x$:
\be
N_i(x)=I\otimes\cdots\otimes|x\rangle\langle x|\otimes\cdots\otimes I
\ee
with $|x\rangle\langle x|$ at the $i$th place and the identity elsewhere.
With $x_1,\ldots,x_N$ in $V$ and $g$ in $S_N$ let
\be
(x_i)=(x_1,x_2,\ldots,x_N)\;\;,\;\;(x_{g(i)})=(x_{g(1)},\ldots,x_{g(N)})\;.
\ee
The vectors
\be\label{3.16}
|(x_i)\rangle=|x_1,\ldots,x_N\rangle
\ee
form an orthonormal basis in $\ell^2(V)^{\otimes N}$. For any subset $C$
of $\{1,2,\ldots,N\}$ and $x$ in $V$ let
\be\label{3.17}
N_C(x)=\sum_{i\in C}N_i(x)\;.
\ee
Any $\ketx$ is an eigenvector of $N_C(x)$ with the eigenvalue
\be\label{3.17a}
n_{C,(x_i)}(x)=\sum_{i\in C}\langle x|x_i\rangle\;.
\ee
For a permutation $g$ let $C_j=C_j(g)$, $j=1,\ldots,k=k(g)$ stand for the
(support of the) cycles of $g$ (see Eq.(\ref{2.7a})). Now $n_{C_j,(x_i)}(x)$ is
a {\em joint cycle-site occupation number} in the configuration $(x_i)$:
it gives the number of particles which are simultaneously at the site $x$
and in the cycle $C_j$. These numbers can be united in a vector
\be
\ngx=(n_{C_1,(x_i)}(x),\ldots,n_{C_k,(x_i)}(x))\;.
\ee
The $\ell^1$-norm of this vector is independent of $g$,
\be
\|\ngx\|_{_1}=\sum_j n_{C_j,(x_i)}(x)=\sum_{i=1}^N\langle x|x_i\rangle
\ee
and gives the number of particles at $x$ in the configuration $(x_i)$.
Now for $x\neq y$ in $V$ the spin pair correlation reads
\bea\label{3.20}
\lefteqn{\sxy}\\
&&=s(s\!+\!1)\frac{\sum_{[p]}\epsilon[p]d^{k[p]}q[p]\sum_{(x_i)}
\gxex\ngxy}{\sumpi\epsilon\crpi d^{k\crpi}q\crpi\sum_{(x_i)}\gxex}\nonumber\;.
\eea
As earlier, $g$ is any element of the class $\crpi$; $\sum_{(x_i)}$ is a
short-hand for $\sum_{x_1}...\sum_{x_N}$. This formula is derived in the
Appendix. One can immediately see that for bosons the pair correlations are
strictly positive. Let us introduce an enlarged event space consisting of the
couples $(g,(x_i))$ and define the probability of $(g,(x_i))$ as
\be\label{3.21}
\pvn(g,(x_i))=\frac{d^{k(g)}}{N!Q_{V,N}}\gxex\;.
\ee
Then for bosons
\be\label{3.22}
\sxy=s(s+1)\evn(\ngxy)\;,
\ee
the expectation value on the right-hand side taken with the probabilities
(\ref{3.21}). Thus, the spin pair correlations are proportional to the density-
density correlations restricted to particles belonging to the same cycle.
The latter is well-defined also for spinless bosons.
In some cases summing over all terms of the numerator of (\ref{3.20}) in which
$\ngxy\geq 2$ yields zero. Then Eq.(\ref{3.20}) can be rewritten more simply as
\be\label{3.23}
\sxy=s(s+1)\frac{\sum_{[p]:p_1>1}\epsilon\crpi d^{k\crpi}q\crpi\sum_{(x_i)}'
\gxex}{\sumpi\epsilon\crpi d^{k\crpi}\sum_{(x_i)}\gxex}\;.
\ee
Here $\sum'$ means that after choosing $g$ in the class $\crpi$ the summation
is restricted to those configurations in which both $x$ and $y$ are singly
occupied and the (labels of the) corresponding two particles belong to the
same cycle of $g$. The restriction $p_1>1$ (cf. Eq.(\ref{1.2})) excludes
the class formed by the unit $e$ of $S_N$: this is necessary because the
inner product in Eq.(\ref{3.20}) vanishes for $g=e$. Equation (\ref{3.23})
is valid for bosons and fermions if the interaction contains a hard-core
repulsion: Doubly occupied sites being excluded, $\ngxy=0$ or $1$ for the
nonvanishing terms.
Equation (\ref{3.23}) is also valid for spin-$1/2$ fermions independently
of the form of the interaction. In this case the sum over the classes
for fixed $(x_i)$ yields zero if $x$ or $y$ is multiply occupied [S1,S2].
This can be understood by noticing that triple and higher encounters are
illicit and a double occupation at, say, $x$ results in a spin singlet on
which $\spin(x)$ gives $0$. The only difference between Eqs.(\ref{1.6}) and
(\ref{3.23}) is that in the latter we have executed the trivial summation
over the permutations within each class.
Let $x\sim y$ denote the event that $x$ and $y$ fall in the same cycle, i.e.,
\be
\{x\sim y\}=\{(g,(x_i)): x_i=x, x_j=y \mbox{\ for some\ } i\sim_g j\}\;.
\ee
For bosons with a hard-core interaction
$\ngxy$ is the indicator function of $x\sim y$,
therefore from (\ref{3.22}) or (\ref{3.23}) we get
\be
\sxy=s(s+1)P(x\sim y)\;.
\ee
For the general boson gas, combining Eqs.(\ref{1.5}), (\ref{3.10}) and
(\ref{3.22}) we obtain
\be\label{3.27}
E(\xi)=\frac{1}{N}\sum_{x,y}E(\ngxy)
\ee
which reduces to
\be\label{3.28}
E(\xi)=\frac{1}{N}\sum_{x,y}P(x\sim y)
\ee
in the case of a hard-core interaction. These formulas hold true whether or
not the bosons have a spin. Equation (\ref{3.27})
can be obtained also directly by noticing that
\be
P(i\sim j)=\sum_{x,y}\sum_{g:i\sim j}\sum_{(x_k):x_i=x,x_j=y}P(g,(x_k))
\ee
and
\be
\sum_{i,j}P(i\sim j)=NE(\xi)\;.
\ee
In contrast to Eq.(\ref{3.28}), the latter equation is always true and follows
from Eq.(\ref{2.24}) with the definition
\be
P(i\sim i)=1\;.
\ee
One may observe a resemblance between $E(\ngxy)$ and the one-particle reduced
density matrix [PO], although the former is more complicated. However, $E(\xi)$
is much easier to compute, at least in the ground state (see Section 5), than
the corresponding sum determining ODLRO ($VA_1$ in [PO]).
\section{Events and probabilities in the infinite system}
\setcounter{equation}{0}
When speaking about a percolation transition one makes allusion to a
phenomenon occurring in an infinite event space. In our case this will be the
family of all the bijections from the set of positive integers onto itself,
which we denote by $\sinfty$. The permutation group $S_N$ can be embedded into
$\sinfty$ by defining $g(i)=i$ for any $g$ in $S_N$ and $i>N$. In this way
\be
S_1\subset S_2\subset\cdots\subset\sinfty\;.
\ee
Clearly, $S_f\equiv\cup_{N=1}^{\infty}S_N$ does not exhaust $\sinfty$. $S_f$
is the family of all the finite permutations, i.e., those having a finite
number of nontrivial cycles. Let $S_0=\sinfty-S_f$. $S_0$ contains bijections
all the cycles of which are finite, an example is $g(2i-1)=2i$, $g(2i)=2i-1$
for every $i\geq 1$. However, "almost all" elements of $S_0$ contain at least
one infinite cycle because $S_0$ is an uncountable set,
while the family of all the bijections in which every number is in a finite
cycle is countable.
The random variables $\nu_j$ and $\xi_i$ introduced in Section 2 are naturally
defined as functions on $\sinfty$ with values in $[0,\infty]$ and $[1,\infty]$,
respectively. The level sets
\bea
B_{in}&=&\{\xi_i=n\}\;,\;\;\;n\geq 1\;,\nonumber\\
B_{i0}&=&\{\xi_i=\infty\}
\eea
are then subsets of $\sinfty$. For any $V$ and $N$ we redefine $\pvn$ as a
probability measure on the $\sigma$-algebra $\cal F(A)$ generated by
${\cal A}=\{B_{im}\}_{i\geq 1,m\geq 1}$ by setting
\be\label{4.2}
\pvn'(\ibim)=\pvn(\ibim\cap S_N)
\ee
and dropping the prime immediately. Notice that
\be
S_N=\cap_{i=N+1}^{\infty}B_{i1}\;.
\ee
We are interested in the limit of (\ref{4.2}) when $N$ and $V$ tend to infinity
and $N/V$ goes to the density $\rho$. By the diagonal process [Rud] one can
choose a subsequence $(V_n,N_n)$ on which these limits simultaneously exist for
every $k$ and $m_1,m_2,\ldots,m_k\geq 1$ finite (thus, for a countable number
of events):
\be\label{4.4}
P_{\rho}(\ibim)=\lim_{n\toinf}P_{V_n,N_n}(\ibim)\;.
\ee
It is obvious that replacing $i_j$ by $g(i_j)$ ($g\in\sinfty$ arbitrary) yields
the same limit. Now
\be\label{4.5}
\sum_{n=1}^{\infty}P_{\rho}(B_{in})\leq 1
\ee
because the inequality holds for finite sums. Therefore one can define the
"probability of cycle percolation" by
\be\label{4.6}
\pro(B_{i0})=1-\sum_{n=1}^{\infty}\pro(B_{in})\;.
\ee
That this is indeed a probability, i.e., that $\pro$ extends from (\ref{4.4})
to a unique probability measure on $\cal F(A)$, remains to be proved. Let us
briefly resume the problem we encounter here. Let $\cal B$ denote the class of
all finite intersections of elements of $\cal A$. The numerical function
$\pro$ is defined on $\cal B$ via Eq.(\ref{4.4}). Furthermore, let $\cal C$
be the class of all finite unions of elements of $\cal A$ and
\be
{\cal D}=\{B-C|B\in{\cal B}, C\in{\cal C}\}\;.
\ee
We call the elements of $\cal D$ "monomials". One can check easily that the
class of all finite unions of monomials is just the ring $\cal R(A)$ generated
by $\cal A$, and the limit $\pro$ of $P_{V_n,N_n}$ exists on $\cal R(A)$ and
defines a {\em content} [Ba]: $0\leq\pro\leq 1$, $\pro(\emptyset)=0$ and
$\pro(\cup_{i=1}^n A_i)=\sum_{i=1}^n\pro(A_i)$ for any finite number of
pairwise disjoint sets $A_1,\ldots,A_n\in\cal R(A)$. If $\pro$ is
$\sigma$-additive on $\cal R(A)$, it has a unique extension by continuity and
complementation as a probability measure on $\cal F(A)$. Proving
$\sigma$-additivity is equivalent to show that if $(A_i)$ is an infinite
sequence in $\cal R(A)$, $A_1\supset A_2\supset\cdots$ and
$\cap_{i=1}^{\infty}A_i=\emptyset$ then $\pro(A_i)\rightarrow 0$. If $A_i$
are monomials, this can be seen to be true by using the inequalities
(\ref{2.36}). The general case is much more complicated and Eq.(\ref{2.36})
may not suffice to get $\sigma$-additivity.
Notice that the
strict inequality in Eq.(\ref{4.5}) in itself indicates a phase transition
independently of whether or not $\pro$ is a probability measure. However,
in the absence of this result one has to be careful with the interpretation
of the limits of probabilities and expectation values.
One may observe that
$\cal F(A)$ is the smallest $\sigma$-algebra on which cycle percolation can
be defined. It does not contain the event $i\sim j$, and some interesting
random variables like the density of an infinite cycle (see next section)
are not $\cal F$-measurable. The systematic extension of $\pro$ could replace
in some respects the $C^*$-algebraic description of infinite Bose systems.
\section{Ground state cycle percolation}
\setcounter{equation}{0}
We consider a system of interacting bosons possibly with a $d$-valued spin
(Eq.(\ref{2.11})) and assume that
the Hamiltonian (\ref{2.2}) has a unique overall
ground state in the spinless Hilbert space ${\cal H}_0^{\otimes N}$.
As discussed in the Introduction, this holds true under very general assumptions
on the interaction, domain and boundary condition. Uniqueness proofs are based
on Perron-Frobenius type theorems ([KR], [GJ]) and show that
the ground state wave function can be chosen
to be positive. Uniqueness and positivity then implies that it is
symmetric under any permutation of the particle positions.
As we see below, this suffices to prove ground state cycle percolation.
Let $E_0=E_0(N)$ denote the energy of the ground state. Then
\bea\label{5.1}
\lim_{\beta\toinf}e^{\beta E_0}\qvn
=N!^{-1}\sum_g d^{k(g)}=\sumpi d^{k\crpi}q\crpi\equiv Q_N(d)\;.
\eea
This limit gives the number of linearly independent ground states of the Bose
gas. Since the degeneracy comes
exclusively from the spins, we have
\bea\label{5.2}
Q_N(d)=\left(\begin{array}{c} d+N-1\\
N \end{array}\right)\;,
\eea
the number of ways we can assign $d$ spin values to $N$ undistinguishable
particles.
The zero temperature limit of the probability distribution for the permutations
is
\be
P_N(g)=d^{k(g)}/N! Q_N(d)\;.
\ee
In particular, for $d=1$ all permutations are equally probable.
We wish to determine $P_N(\xi_i=j)$. This goes via the determination of
$E_N(\nu_j)$, see Eq.(\ref{2.21}), which can be done exactly. However, we
start with an approximate method which will be reapplied for the free Bose gas
at positive temperatures [S3]. From Eqs.(\ref{2.9}) and (\ref{5.1}) for any
$\lambda$ we find
\be
P_N((\nu_k)\!=\!(n_k))=e^{-\lambda N}Q_N(d)^{-1}\prod_{k=1}^N
\left(\frac{de^{\lambda k}}{k}\right)^{n_k}\frac{1}{n_k!}
\ee
if $\sum kn_k=N$ and $0$ otherwise. Let us forget about the constraint
(\ref{2.8}) for a moment. Then the $\nu_j$ are independent
Poisson-distributed random variables with mean value
\be\label{5.5}
E(\nu_j)=de^{\lambda j}/j\;.
\ee
Now let us take the constraint into account in average, i.e., choose
$\lambda$ so that
\be\label{5.6}
\sum_{k=1}^N kE(\nu_k)=d\sum_{k=1}^N e^{\lambda k}=N\;.
\ee
One can check that $E(\nu_j)$ determined from (\ref{5.5}), (\ref{5.6}) is the
most probable value of $\nu_j$ obtained as a conditional extremum in a
continuous approximation. The Lagrange multiplier $\lambda$ can be identified
with the chemical potential. In particular, for $d=1$ we get $\lambda=0$ and
\be\label{5.7}
E(\nu_j)=1/j\;.
\ee
It turns out that this is the exact result. Let us compute $E_N(\nu_j)$ by
fully respecting the constraint (\ref{2.8}). With the notations of
Eq.(\ref{2.32}) the result is
\bea\label{5.8}
E_N(\nu_j)&=&\sum_m mP_N(\nu_j=m)
=\sum_{m\geq 1}m\sum_{\crpi:n_j=m}d^{k\crpi}q\crpi/Q_N(d)\nonumber\\
&=&\frac{d}{j}\sum_{\crpi_{N-j}}d^{k\crpi_{N-j}}q\crpi_{N-j}/Q_N(d)
=\frac{d}{j}Q_{N-j}(d)/Q_N(d)
\eea
which is equal to (\ref{5.7}) for $d=1$. For $d>1$, Eq.(\ref{5.5}) gives a
reasonably good approximation if $j\ll N$. To see this, notice that in this
case Eq.(\ref{5.8}) yields
\be
E_N(\nu_j)\approx \frac{d}{j}e^{-\frac{d-1}{N}j}\;,
\ee
while the solution of Eq.(\ref{5.6}) for $\lambda$ is
\be
\lambda=-a/N
\ee
with $d\!-\!1From Eqs.(\ref{2.21}) and (\ref{5.8})
\be\label{5.11}
P_N(\xi_i=j)=\frac{d}{N}Q_{N-j}(d)/Q_N(d)\;.
\ee
For $d=1$ all the cycle lengths are equally probable. Moreover,
\be
p(n)\equiv\pro(B_{in})=\lim_{N\toinf}P_N(\xi_i=n)=0
\ee
for any $n\geq 1$. Thus the percolation probability $1-\sum_{n\geq 1} p(n)=1$.
Furthermore, applying Eq.(\ref{5.2}) and the Pascal triangle identity
\bea\label{5.13}
\sum_{k=0}^m\left(\begin{array}{c} n+k\\n\end{array}\right)
=\left(\begin{array}{c}n+m+1\\n+1\end{array}\right)
\eea
we obtain
\be\label{5.14}
E_N(\xi_i)=\frac{N+d}{d+1}\;.
\ee
>From Eq.(\ref{5.11}) we can infer the limit distribution of the cycle densities.
Let $0\leq a**From (\ref{5.2}) and (\ref{5.8}) one obtains
\be
\lim_{N\toinf}E_N(\sum_{j\geq xN}\nu_j)\geq d\ln\frac{1}{x}+d(d-1)(1-x)
\ee
(with equality at $d=1$) which diverges as $x$ goes to zero. In this respect,
cycle percolation resembles bond percolation on a tree where the number of
infinite clusters is infinite when percolation takes place.
A physical consequence of the existence of macroscopic cycles is that the
ground state of a system of spinning bosons is ferromagnetic.
>From Eqs.(\ref{3.10}) and (\ref{5.14}) the ground state magnetic susceptibility
is
\be
\lim_{\beta\toinf}\beta^{-1}\chi_N=\frac{1}{3}s(s+1)\frac{N+2s+1}{2s+2}
=s(N+2s+1)/6\;.
\ee
Comparing with Eq.(\ref{1.5}) or (\ref{3.12}), we see that for $s>1/2$ the
magnetic moment cannot saturate: $S_{\max}=Ns$ while
\be
\frac{\langle\spin^2\rangle}{S_{\max}(S_{\max}+1)}\gobgo
\frac{1}{2}\frac{N+2s+1}{Ns+1}<1
\ee
if $N>1$. The reason is that there exist ground states with spin quantum number
less than $Ns$, and these are all mixed together when the temperature goes
to zero in zero magnetic field.
\section{Percolation transition in the free Bose gas}
\setcounter{equation}{0}
For the free Bose gas
\be
H_N=\sum_{i=1}^N H_i^0\;\;,\;\;H_i^0=-\frac{\hbar^2}{2m}\Delta_i
\ee
and the trace in Eq.(\ref{2.14}) factorizes according to the cycles: If $g$
has $n_k$ cycles of length $k$,
\be\label{6.2}
\Tr_{{\cal H}_0^{\otimes N}}U_0(g)\ebh=\prod_{k=1}^N\left(\tr e^{-\beta kH^0}
\right)^{n_k}
\ee
where $\tr=\Tr_{{\cal H}_0}$. For the probability of a conjugacy class
we find
\be
\pvn((\nu_k)\!=\!(n_k))
=\frac{1}{\qvn}\prod_{k=1}^N\left(\frac{d\tr e^{-\beta kH^0}}{k}\right)^
{n_k}\frac{1}{n_k!}
\ee
if $\sum kn_k=N$ and zero otherwise. This yields, similarly to Eqs.(\ref{5.8})
and (\ref{5.11}),
\be
\evn(\nu_k)=\frac{d}{k}\tr e^{-\beta kH^0}Q_{V,N-k}/\qvn
\ee
and
\be
\pvn(\xi_i=k)=\frac{d}{N}\tr e^{-\beta kH^0}Q_{V,N-k}/\qvn\;.
\ee
For the sake of simplicity, let $V$ be a $D$-dimensional hypercube of side
$L$ and choose periodic boundary condition. Then due to the zero eigenvalue
of $H^0$
\be\label{6.6}
\qvn>Q_{V,N-1}\;.
\ee
On the other hand, by simple computation
\be\label{6.7}
\tr e^{-\beta kH^0}\leq \left[1+\frac{L}{B\sqrt{k}}\right]^D
\ee
where $B=\hbar(2\pi\beta/m)^{1/2}$. Thus,
\bea\label{6.8}
\pro(\xi_i=k)=\lim_{V,N\toinf}\pvn(\xi_i=k)\leq\frac{d}{\rho B^D}k^{-D/2}
\eea
and therefore
\be\label{6.9}
\sum_{k=1}^{\infty}\pro(\xi_i=k)<1 \mbox{\ \ if\ \ }
\rho B^D>d\sum_{k=1}^{\infty}k^{-D/2}
\ee
which can be satisfied if $D>2$.
In Eq.(\ref{6.8}) and below the limit is taken so that $N/V$ goes to $\rho$.
Notice that the second inequality in (\ref{6.9}) is the well-known condition
for the Bose-Einstein condensation (see e.g. [Hu] in the case $d=1$ and
[CL] for $d>1$). So the probability of cycle percolation is positive when the
condition for the condensation is satisfied. To complete the discussion we
should prove the converse statement:
\be\label{6.10}
\sum_{k=1}^{\infty}\pro(\xi_i=k)=1 \mbox{\ \ if\ \ }
\rho B^DN$ (cf. Eq.(\ref{4.2})) and
$\tr e^{-\beta kH^0}/N$ can be bounded by a constant.
To get such an $f(k)$ observe that, if in (\ref{6.10}) the inequality holds
true, there is no condensation and in the thermodynamic limit the canonical
and grand-canonical ensembles are equivalent ([Rue], Thm.3.5.8). For fixed
$\beta$ let $z(\rho)<1$ be the fugacity corresponding to the density $\rho$
and define
\be
z_{V,N}=Q_{V,N-1}/\qvn\;.
\ee
By adapting the proof of Van der Linden obtained for classical systems [VdL],
one can show that $z_{V,N}$ converges to $z(\rho)$ which is continuous,
increasing and smaller than $1$ for $\rho<\rho_c(\beta)$. This implies that
\be
\lim_{V,N\toinf}Q_{V,N-k}/\qvn=z(\rho)^k
\ee
for all $k$. Here we would need the uniformity of the convergence and,
in fact, it is plausible to conjecture that for any $\rho_0<\rho_c(\beta)$
and $\varepsilon>0$ one can find a $V_0<\infty$ such that
\bea\label{6.14}
\left|\frac{Q_{V,M-1}}{Q_{V,M}}-z\left(\frac{M}{V}\right)\right|
<\varepsilon
\eea
if $V>V_0$ and $0**