\documentstyle[12pt] {article}
\setlength{\baselineskip}{2.ex}
\renewcommand{\baselinestretch}{1.5}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\begin{document}
\begin{center}
{\large \bf Quantum Chaos in a Yang--Mills--Higgs System}
\end{center}
\vskip 1. truecm
\begin{center}
{\bf Luca Salasnich}
\footnote{E--Mail: salasnich@padova.infn.it}
\vskip 0.5 truecm
Dipartimento di Matematica Pura ed Applicata,\\
Universit\`a di Padova, Via Belzoni 7, I 35131 Padova, Italy\\
and\\
Istituto Nazionale di Fisica Nucleare, Sezione di Padova, \\
Via Marzolo 8, I 35131 Padova, Italy \\
\end{center}
\vskip 1. truecm
\begin{center}
{\bf Abstract}
\end{center}
\vskip 0.5 truecm
\par
We study the energy fluctuations
of a spatially homogeneous SU(2) Yang--Mills--Higgs system.
In particular, we analyze the nearest--neighbour spacing
distribution which shows
a Wigner--Poisson transition by increasing the value of the Higgs
field in the vacuum. This transition is a clear quantum signature
of the classical chaos--order transition of the system.
\vskip 0.5 truecm
\begin{center}
To be published in Modern Physics Letters A
\end{center}
\newpage
\section{Introduction}
\par
In the last years there has been much interest in classical chaos
in field theories. It is now well known that the spatially uniform limits
of scalar electrodynamics and Yang--Mills theory exhibit classical
chaotic motion$^{1)-8)}$. On the other hand, in field theories,
less attention has been paid to {\it quantum chaos},
i.e. the study of properties of quantum systems
which are classically chaotic$^{9)}$.
\par
The energy fluctuation properties of systems with underlying
classical chaotic behaviour and time--reversal symmetry agree with
the predictions of the Gaussian Orthogonal Ensemble (GOE) of
random matrix theory, whereas quantum analogs of classically
integrable systems display the characteristics
of the Poisson statistics$^{9)-12)}$.
Some results in this direction for field theories have been obtained by
Halasz and Verbaarschot: they studied the QCD lattice spectra
for staggered fermions and its connection to random matrix theory$^{13)}$.
\par
In this paper we study quantum chaos in a field--theory schematic model.
We analyze the energy fluctuation properties
of the spatially homogeneous SU(2) Yang--Mills--Higgs (YMH) system
(see Ref. 1--4). We show that these fluctuations give a clear quantum signature
of the classical chaos--order transition of the system.
\par
The Lagrangian density of the SU(2) YMH system$^{14)}$ is given by
\beq
L={1\over 2}(D_{\mu}\phi )^+(D^{\mu}\phi ) -V(\phi )
-{1\over 4}F_{\mu \nu}^{a}F^{\mu \nu a} \; ,
\eeq
where
\beq
(D_{\mu}\phi )=\partial_{\mu}\phi - i g A_{\mu}^b T^b\phi
\; ,
\eeq
\beq
F_{\mu \nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+
g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c} \; ,
\eeq
with $T^b=\sigma^b/2$, $b=1,2,3$, generators of the SU(2) algebra,
and where the potential of the scalar field (the Higgs field) is
\beq
V(\phi )=\mu^2 |\phi|^2 + \lambda |\phi|^4 \; .
\eeq
We work in the (2+1)--dimensional Minkowski space ($\mu =0,1,2$) and
choose spatially homogeneous Yang--Mills and the Higgs fields
\beq
\partial_i A^a_{\mu} = \partial_i \phi = 0 \; , \;\;\;\; i=1,2
\eeq
i.e. we consider the system in the region in which space fluctuations of
fields are negligible compared to their time fluctuations.
\par
In the gauge $A^a_0=0$ and using the real triplet representation for the
Higgs field we obtain
$$
L={\dot{\vec \phi}}^2 +
{1\over 2}({\dot {\vec A}}_1^2+{\dot {\vec A}}_2^2)
-g^2 [{1\over 2}{\vec A}_1^2 {\vec A}_2^2
-{1\over 2} ({\vec A}_1 \cdot {\vec A}_2)^2+
$$
\beq
+({\vec A}_1^2+{\vec A}_2^2){\vec \phi}^2
-({\vec A}_1\cdot {\vec \phi})^2 -({\vec A}_2 \cdot {\vec \phi})^2 ]
-V( {\vec \phi} ) \; ,
\eeq
where ${\vec \phi}=(\phi^1,\phi^2,\phi^3)$,
${\vec A}_1=(A_1^1,A_1^2,A_1^3)$ and ${\vec A}_2=(A_2^1,A_2^2,A_2^3)$.
\par
When $\mu^2 >0$ the potential $V$ has a minimum at $|{\vec \phi}|=0$,
but for $\mu^2 <0$ the minimum is at
$$
|{\vec \phi}_0|=\sqrt{-\mu^2\over 4\lambda }=v \; ,
$$
which is the non zero Higgs vacuum. This vacuum is degenerate
and after spontaneous symmetry breaking the physical vacuum can be
chosen ${\vec \phi}_0 =(0,0,v)$. If $A_1^1=q_1$, $A_2^2=q_2$
and the other components of the Yang--Mills fields are zero,
in the Higgs vacuum the Hamiltonian of the system reads
\beq
H={1\over 2}(p_1^2+p_2^2)
+g^2v^2(q_1^2+q_2^2)+{1\over 2}g^2 q_1^2 q_2^2 \; ,
\eeq
where $p_1={\dot q_1}$ and $p_2={\dot q_2}$. Here $w^2=2 g^2v^2$ is the
mass term of the Yang--Mills fields. This YMH Hamiltonian is
a toy model for classical non--linear dynamics, with the attractive feature
that the model emerges from particle physics. In the next sections we
analyze first the classical chaos--order transition of the YMH system
and then its connection to the quantal fluctuations of the energy levels.
\section{Classical chaos--order transition}
\par
A classical chaos--order transition for the YMH system
has been observed previously by different authors:
Savvidy used the Chirikov'criterion$^{1)}$, Kawabe and Ohta studied
the Lyapunov exponents$^{3)}$ and Salasnich analyzed the
quantal overlapping resonances$^{4)}$.
In this paper we study the chaotic behaviour of this YMH system by using
the Gaussian curvature criterion of the
potential energy$^{16)}$ and the Poincar\`e Sections$^{17)}$.
\par
At low energy the motion near
the minimum of the potential
\beq
V(q_1,q_2)=g^2 v^2 (q_1^2+q_2^2)+{1\over 2} g^2 q_1^2 q_2^2 \; ,
\eeq
where the Gaussian curvature is positive, is periodic or quasiperiodic and is
separated from the instability region by a line of zero curvature;
if the energy is increased, the system will be for some initial conditions
in a region of negative curvature, where the motion is chaotic.
According to this scenario, the energy $E_c$ of chaos--order transition
is equal to the minimum value of the line of zero gaussian
curvature $K(q_1 ,q_2 )$ on the potential--energy surface.
For our potential the gaussian curvature vanishes at the points
that satisfy the equation
\beq
{\partial^2 V \over \partial q_1^2}
{\partial^2 V \over \partial q_2^2}-
({\partial^2 V \over \partial q_1 \partial q_2})^2=
(2g^2v^2 +g^2 q_2^2)(2g^2v^2+g^2q_1^2)-4g^4 q_1^2q_2^2=0 \; .
\eeq
It is easy to show that the minimal energy on the
zero--curvature line is given by:
\beq
E_c=V_{min}(K=0,\bar{q_1})=6 g^2 v^4 \; ,
\eeq
and by inverting this equation
we obtain $v_c=(E /6g^2)^{1/4}$. We conclude that
there is a order--chaos transition by increasing the energy $E$
of the system and a chaos--order transition by increasing
the value $v$ of the Higgs field in the vacuum (see also Ref. 2). Thus,
there is only one transition regulated by the unique parameter $E/(g^2v^4)$.
\par
It is important to point out that
{\it in general} the curvature
criterion guarantees only a {\it local instability}$^{16)}$
and should therefore
be combined with the Poincar\`e sections$^{17)}$ (see Ref. 18).
The classical equations of motion of the YMH system are
\beq
{\dot q_1}=p_1 \; , \;\;\;\;
{\dot q_2}=p_2 \; , \;\;\;\;
{\dot p_1}=-2g^2v^2 q_1 - g^2 q_1 q_2^2 \; , \;\;\;\;
{\dot p_2}=-2g^2v^2 q_2 - g^2 q_1^2 q_2 \; .
\eeq
We use a fourth--order Runge--Kutta
method$^{19)}$ to compute the classical trajectories.
The conservation of energy restricts any trajectory of the four--dimensional
phase space to a three--dimensional energy shell. At a particular energy
the restriction $q_1=0$ defines a two--dimensional surface in
the phase space, which is called Poincar\`e section.
Each time a particular trajectory passes through the surface
a point is plotted at the position of intersection $(q_2,p_2)$.
We employ a first--order interpolation process to reduce the inaccuracies
due to the use of a finite step length$^{17)}$.
\par
In Figure 1 we plot the Poincar\`e sections for different values of
the Higgs vacuum $v$ but with the same energy $E$ and
interaction $g$. Chaotic regions on the surface of section
are characterized by a set of randomly distributed points
and regular regions by dotted or solid curves.
The pictures show that the parameter $v$ plays
an important role: for large values it makes the system regular.
In fact, if we increase the harmonic part of the YMH potential
the effect of the nonlinear term becomes less important.
These numerical calculations confirm the analytical
predictions of the curvature criterion: with $E =10$ and
$g=1$ we get the critical value of the onset of chaos
$v_c=(E /6g^2)^{1/4}\simeq 1.14$,
in very good agreement with the Poincar\`e sections.
\section{Quantum signature of the chaos--order transition}
\par
In quantum mechanics the generalized coordinates of the YMH system
satisfy the usual commutation rules $[{\hat q}_k,{\hat p}_l]=i\delta_{kl}$,
with $k,l=1,2$. Introducing the creation and destruction operators
\beq
{\hat a}_k=\sqrt{\omega \over 2}{\hat q}_k +
i \sqrt{1\over 2\omega}{\hat p}_k \; ,
\;\;\;\;
{\hat a}_k^+ = \sqrt{\omega \over 2}{\hat q}_k -
i \sqrt{1\over 2\omega}{\hat p}_k \; ,
\eeq
the quantum YMH Hamiltonian can be written$^{15)}$
\beq
{\hat H}={\hat H}_0 + {1\over 2} g^2 {\hat V} \; ,
\eeq
where
\beq
{\hat H}_0= \omega ({\hat a}_1^+ {\hat a}_1 + {\hat a}_2^+ {\hat a}_2 + 1) \; ,
\eeq
\beq
{\hat V}= {1 \over 4 \omega^2} ({\hat a}_1 +{\hat a}_1^+)^2
({\hat a}_2 +{\hat a}_2^+)^2 \; ,
\eeq
with $\omega^2 = 2 g^2 v^2$ and $[{\hat a}_k,{\hat a}_l^+] = \delta_{kl}$,
$k,l=1,2$.
\par
The most used quantity to study the local fluctuations of the energy levels
is the spectral statistics $P(s)$. $P(s)$ is
the distribution of nearest--neighbour spacings
$s_i=({\tilde E}_{i+1}-{\tilde E}_i)$
of the unfolded levels ${\tilde E}_i$.
It is obtained by accumulating the number of spacings that lie within
the bin $(s,s+\Delta s)$ and then normalizing $P(s)$ to unity$^{9)-12)}$.
\par
For quantum systems whose classical analogs are integrable,
$P(s)$ is expected to follow the Poisson limit, i.e.
$P(s)=\exp{(-s)}$. On the other hand,
quantal analogs of chaotic systems exhibit the spectral properties of
GOE with $P(s)= (\pi / 2) s \exp{(-{\pi \over 4}s^2)}$, which is the
so--called Wigner distribution$^{9)-12)}$.
The distribution $P(s)$ is the best spectral statistics to analyze
shorter series of energy levels and
the intermediate regions between order and chaos.
\par
Seligman, Verbaarschot and Zirnbauer$^{20)}$ analyzed
a class of two--dimensional anharmonic oscillators with
polynomial perturbation by using the Brody distribution$^{21)}$
\beq
P(s,\omega)=\alpha (\omega +1) s^{\omega} \exp{(-\alpha s^{\omega+1})} \; ,
\eeq
with
\beq
\alpha = \big( \Gamma [{\omega +2\over \omega+1}] \big)^{\omega +1} \; .
\eeq
This distribution interpolates between the Poisson distribution ($\omega =0$)
of integrable systems and the Wigner distribution ($\omega =1$) of
chaotic ones, and thus the parameter $\omega$ can be used as a simple
quantitative measure of the degree of chaoticity.
\par
We compute the energy levels $\{ E_i \}$ with
a numerical diagonalization of the truncated matrix of the quantum
YMH Hamiltonian in the basis of the harmonic oscillators$^{22)}$.
If $|n_1 n_2>$ is the basis of the occupation numbers of the two
harmonic oscillators, the matrix elements are
\beq
= \omega (n_1+n_2+1)
\delta_{n_{1}^{'}n_{1}} \delta_{n_{2}^{'}n_{2}} \; ,
\eeq
and
$$
=
{1 \over 4 \omega^2}
[\sqrt{n_{1}(n_{1}-1)} \delta_{n^{'}_{1}n_{1}-2}
+\sqrt{(n_{1}+1)(n_{1}+2)}\delta_{n^{'}_{1}n_{1}+2}+
(2n_{1}+1)\delta_{n^{'}_{1}n_{1}}]\times
$$
\beq
\times[\sqrt{n_2 (n_2-1)}\delta_{n^{'}_2 n_2-2}+ \sqrt{(n_2+1)(n_2+2)}
\delta_{n^{'}_2 n_2+2}+ (2n_2+1)\delta_{n^{'}_2 n_2}] \; .
\eeq
The symmetry of the potential enables us to split
the Hamiltonian matrix into 4 sub--matrices
reducing the computer storage required. These sub--matrices are related
to the parity of the two occupation numbers $n_1$ and $n_2$:
even--even, odd--odd, even--odd, odd--even.
The numerical energy levels depend on the dimension of the truncated matrix:
we compute the numerical levels in double precision
increasing the matrix dimension until the first 100 levels converge
within $8$ digits (matrix dimension $1156\times 1156$)$^{22),23)}$.
\par
We use the first $100$ energy levels of the 4 sub--matrices
to calculate the $P(s)$ distribution.
In order to remove the secular variation of the level density as a function
of the energy $E$, for each value of the coupling constant the
corresponding spectrum is mapped, by a numerical procedure described in
Ref. 24, into one which has a constant level density:
$\{ E_i \} \to \{ {\tilde E_i} \}$ (unfolding procedure).
We use the following standard procedure to avoid mixing
between states of different symmetry classes:
1) the diagonalization is performed for each sub-matrix
(first 100 levels for each sub-matrix);
2) the unfolding is done for each sub-matrix;
3) the spacings are calculated for each sub-matrix;
4) the spacings of the 4 sub-matrices are
accumulated to plot the P(s) distribution.
\par
In Figure 2 we plot the $P(s)$ distribution
for different values of the parameter $v$.
The figure shows a Wigner--Poisson transition by increasing the value $v$
of the Higgs field in the vacuum.
By using the P(s) distribution and the Brody function
it is possible to give a quantitative measure
of the degree of quantal chaoticity of the system.
Our numerical calculations show clearly the quantum
chaos--order transition and its connection to the classical one.
\section{Conclusions}
\par
The chaotic behaviour of an
homogenous YMH system has been studied both in classical and quantum
mechanics. The Gaussian curvature criterion and the Poincar\`e
sections show that the chaotic behaviour
is regulated by the unique parameter $E/(g^2v^4)$.
The YMH system has a order--chaos transition by
increasing the energy $E$ and a chaos--order
transition by increasing the value $v$ of the Higgs field in the vacuum.
\par
The nearest--neighbour spacing distribution of the energy
levels confirms with great accuracy
the classical chaos--order transition of the YMH system.
In particular, the Brody function shows a Wigner--Poisson transition
for the $P(s)$ distribution in correspondence to the classical
chaos--order transition.
\par
We observe that, as stressed previously, our YMH system is a toy model
but it is very useful because it is possible
to compare classical to quantum chaos.
In the future will be important to study classical
and quantum chaos in more realistic field theories.
\section*{Acknowledgments}
The author is grateful to G. Benettin, V.R. Manfredi, M. Robnik
and A. Vicini for stimulating discussions.
\newpage
\parindent=0.pt
\section*{Figure Captions}
\vspace{0.6 cm}
{\bf Figure 1}: The Poincar\`e sections of the model. From the top:
$v=1$, $v=1.1$ and $v=1.2$. Energy $E = 10$ and interaction $g=1$.
{\bf Figure 2}: $P(s)$ distribution. From the top:
$v=1$ ($\omega=0.92$), $v=1.1$ ($\omega =0.34$) and $v=1.2$ ($\omega =0.01$),
where $\omega$ is the Brody parameter. First 100 energy levels
and interaction $g=1$. The dotted, dashed and solid curves stand
for Wigner, Poisson and Brody distributions, respectively.
\newpage
\section*{References}
\begin{description}
\item{\ 1.} G.K. Savvidy, Nucl. Phys. {\bf B 246}, 302 (1984).
\item{\ 2.} A. Gorski, Acta Phys. Pol. {\bf B 15}, 465 (1984).
\item{\ 3.} T. Kawabe and S. Ohta, Phys. Rev. {\bf D 44}, 1274 (1991).
\item{\ 4.} L. Salasnich, Phys. Rev. {\bf D 52}, 6189 (1995).
\item{\ 5.} T. Kawabe, Phys. Lett. {\bf B 343}, 254 (1995).
\item{\ 6.} L. Salasnich, Mod. Phys. Lett. {\bf A 10}, 3119 (1995).
\item{\ 7.} J. Segar and M.S. Sriram, Phys. Rev. {\bf D 53}, 3976 (1996).
\item{\ 8.} S.G. Matinyan and B. Muller,
Phys. Rev. Lett. {\bf 78}, 2515 (1997).
\item{\ 9.} M.C. Gutzwiller, {\it Chaos in Classical and Quantum Mechanics}
(Springer, Berlin, 1990).
\item{\ 10.} A.M. Ozorio de Almeida, {\it Hamiltonian Systems: Chaos and
Quantization} (Cambridge University Press, Cambridge, 1990).
\item{\ 11.} K. Nakamura, {\it Quantum Chaos}
(Cambridge Nonlinear Science Series, Cambridge, 1993).
\item{\ 12.} G. Casati and B.V. Chirikov, {\it Quantum Chaos}
(Cambridge University Press, Cambridge, 1995).
\item{\ 13.} M.A. Halasz and
J.J.M. Verbaarschot, Phys. Rev. Lett. {\bf 74}, 3920 (1995).
\item{\ 14.} C. Itzykson and J.B. Zuber,
{\it Quantum Field Theory} (McGraw--Hill, New York, 1985).
\item{\ 15.} G.K. Savvidy, Phys. Lett. {\bf B 159}, 325 (1985).
\item{\ 16.} M. Toda, Phys. Lett. {\bf A 48}, 335 (1974).
\item{\ 17.} M. Henon, Physica {\bf D 5}, 412 (1982).
\item{\ 18.} G. Benettin, R. Brambilla and
L. Galgani, Physica {\bf A 87}, 381 (1977).
\item{\ 19.} Subroutine D02BAF, The NAG Fortran Library, Mark 14
(NAG Ltd, Oxford, 1990).
\item{\ 20.} T.H. Seligman, J.J.M. Verbaarschot and M.R. Zirnbauer,
Phys. Rev. lett. {\bf 53}, 215 (1984).
\item{\ 21.} T.A. Brody, Lett. Nuovo Cimento {\bf 7}, 482 (1973).
\item{\ 22.} S. Graffi, V.R. Manfredi and L. Salasnich,
Mod. Phys. Lett. {\bf B 9}, 747 (1995).
\item{\ 23.} Subroutine F02AAF, The NAG Fortran Library, Mark 14
(NAG Ltd, Oxford, 1990).
\item{\ 24.} V.R. Manfredi, Lett. Nuovo Cimento {\bf 40}, 135 (1984).
\end{description}
\end{document}