Symmetry Breaking Study with Deformed Ensembles

TL;DR

This paper introduces a random matrix model to analyze symmetry breaking in physical systems, specifically applying it to eigenfrequency data of an anisotropic quartz block to identify intrinsic symmetries.

Contribution

It develops a new random matrix framework for symmetry breaking and demonstrates its application to real-world vibrational data, aiding symmetry detection.

Findings

Effective modeling of m-fold symmetry coupling

Application to eigenfrequency data of quartz

Potential to identify intrinsic symmetries

Abstract

A random matrix model to describe the coupling of m-fold symmetry in constructed. The particular threefold case is used to analyze data on eigenfrequencies of elastomechanical vibration of an anisotropic quartz block. It is suggested that such experimental/theoretical study may supply powerful means to discern intrinsic symmetries in physical systems.

Equations

$$H=H_0+\lambda H_1,$$

$$P_i=\sum _{j\in I_i}\mid j><j\mid ,$$

$$H_0=\sum _{i=1}^m{P}_i{H}^{GOE}{P}_i$$

$$H_1=\sum _{i=1}^m{P}_i{H}^{GOE}{Q}_i$$

$$P(H,\alpha ,\beta )=Z_N^{-1}\exp \left(-\alpha tr{H}^2-\beta tr{H}_1^2\right)$$

$$\lambda =(1+\frac{\beta }{\alpha }{)}^{-1/2}.$$

$$\rho (E)=\sum _{i=1}^m\frac{M_i}{N}{\rho }_i(E)$$

$$\rho_{i}(E)=\left\{\begin{array}[]{rl}\frac{2}{\pi a_{i}^{2}}\sqrt{{a_{i}^{2}-E^{2}}},&\mid E\mid\leq a\\ 0,&\mid E\mid>a\end{array}\right.$$

$$a_i^2=a^2\left[\frac{M_i}{N}+{\lambda }^2\left(1-\frac{M_i}{N}\right)\right].$$

$$P_{2\times 2}(s,\beta )=\alpha s\exp \left(-\frac{\alpha }{2}{s}^2\right)\sqrt{1+\frac{\beta }{\alpha }}{I}_0(\frac{\beta s^2}{4})\exp \left(-\frac{\beta s^2}{4}\right),$$

$$I_0(x)\to \frac{e^x}{\sqrt{2\pi x}}.$$

$$P_{2\times 2}(s,\beta \to 0)\approx \frac{\pi }{2}exp\left(-\frac{\pi }{4}{s}^2\right)s$$

$${\Delta }_3^g(L)=g\frac{L}{15}+(1-g{)}^2{\Delta }_3\left(\frac{L}{1-g}\right).$$

Full text

The standard ensembles of Random Matrix Theory (RMT) Meht have had wide application in the description of the statistical properties of eigenvalues and eigenfunctions of complex many-body systems. Other ensembles have also been introduced Dyson , in order to cover situations that depart from universality classes of RMT. One such class of ensembles is the so-called Deformed Gaussian Orthogonal Ensemble (DGOE) Pato1 ; Pato2 ; Pato3 ; Carneiro:1991 that proved to be particularly useful when one wants to study the breaking of a discrete symmetry in a many-body system such as the atomic nucleus.

In fact, the use of spectral statistics as a probe of symmetries in physical systems has been a subject of intensive experimental and theoretical investigation following the pioneering work of Bohigas, Giannoni and Schmit Bohigas which showed that the quantal behaviour of classically chaotic systems exhibits the predictions supplied by the RMT. Examples of symmetry breaking in physical systems that have been studied include nuclei Mitch0 ; Mitch , atoms Simons ; Welch and mesoscopic devices such as quantum dots Alhassid .

In the case of nuclei, the Mitchell group at the Triangle Universities Nuclear Laboratory Mitch0 ; Mitch , studied the effect of isospin symmetry breaking, in odd-odd nuclei such as ${}^{26}Al$. They detected the breakdown of this important symmetry by the applications of two statistics: the short-range, nearest neighbor level spacing distribution (NND) and the long range Dyson’s $\Delta$-statistics Mitch0 ; Mitch . These results were well described by a DGOE in which a pair of diagonal blocks is coupled. The strength of the coupling needed to account for the symmetry breaking can be traced to the average matrix element of the Coulomb interaction responsible for this discrete symmetry breaking Pato2 ; Guhr . The justification for the use of block matrices to describe the statistics of a superposition of $R$ spectra with different values of the conserved quatum number can be traced to Refs. Meht ; NovaR . In the case of non-interacting spectra, i.e. if the quantum number is exactly conserved, the answer is a superposition of the $R$ spectra. Since the level repulsion is present in each one of the $R$ spectra, their superposition does not show this feature. Thus, we can say that for each spectra of states of a given value of the quantum number, one attaches a random matrix (GOE). For $R$ spectra each of which has a given value of the conserved quantum number, one would have an $R\times R$ block diagonal matrix. Each block matrix will have a dimension dictated by the number state of that spectra. If the quantum number is not conserved then the $R\times R$ block matrix acquires non-diagonal matrices that measure the degree of the breaking of the associated symmetry. This idea was employed by Guhr and Weidenmüller Guhr and Hussein and Pato Pato1 to discuss isopin violation in the nucleus 26**Al. In reference Pato1 , the random block matrix model was called the Deformed Gaussian Orthogonal Ensemble (DGOE).

In order to study transitions amongst universal classes of ensembles such as order-chaos (Poisson$\rightarrow$GOE), symmetry violation transitions (2GOE$\rightarrow$1GOE), experiments on physical systems are more complicated due to the difficulty of tuning the interaction (except, e.g. in highly excited atoms where the application of a magnetic field allows the study of GOE-GUE transitions). To simulate the microscopic physical systems, one relies on analog computers such as microwave cavities, pioneered by A. Richter and collaborators achim1 and acoustic resonators of Ellegaard and collaboratorsElleg0 ; Elleg ; Elleg2 . It is worth mentioning at this point that the first to draw attention to the applicability of RMT to accoustic waves in physical system was Weaver Weaver .

In the experiment of Ellegaard et al. Elleg what was measured were eigenfrequencies of the elastomechanical vibrations of an anisotropic crystal block with a D3 point-group symmetry. The rectangular crystal block employed by Ellegard was so prepared as to have only a two-fold flip symmetry retained. Then, to all effects, the quartz specimen resembles a system of two three-dimensional Sinai billiards. The statistical treatment of the eigenfrequencies of such a block would follow that of the superposition of two uncoupled GOE’s.

Then, by removing octants of progressively larger radius from a corner of the crystal block this remnant two-fold symmetry was gradually broken. The spectral statistics show a transition towards fully a chaotic system as the octant radius increases. What was then seen was that the measured NND is compatible with a two block DGOE description but the $\Delta$-statistics was discrepant. This discrepancy was attributed to pseudo integrable behavior and this explanation was later implemented with the result that the long-range behavior was fitted at the cost, however, of loosing the previous agreement shown by the NNDAbul .

Here we reanalyse this experiment following the simpler idea of extending the DGOE matrix model Pato3 to consider the coupling of three instead of two GOE’s Carneiro:1991 . We show that within this extension both, the short- and the long-range statistics, are reasonably fitted suggesting that the assumption of the reduction of the complex symmetries of anisotropic quartz block may not be correct. Our findings have the potential of supplying very precise means of testing details of symmetry breaking in pysical systems.

To define the ensembles of random matrices we are going to work with, we recall the construction based on the Maximum Entropy Principle Pato1 , that leads to a random Hamiltonian which can be cast into the form

[TABLE]

where the block diagonal $H_{0}$ is a matrix made of $m$ uncoupled GOE blocks and $\lambda$ ($0\leq\lambda\leq 1)$ is the parameter that controls the coupling among the blocks represented by the $H_{1}$ off-diagonal blocks. For $\lambda=1,$ the $H_{1}$ part completes the $H_{0}$ part and $H=H^{GOE}.$

These two matrices $H_{0}$ and $H_{1}$ are better expressed introducing the following $m$ projection operators

[TABLE]

where $I_{i}$ defines the domain of variation of the row and column indexes associated with $i$th diagonal block of size $M_{i}.$ Since we are specifically interested in the transition from a set of $m$ uncoupled GOE’s to a single GOE, we use the above projectors to generalize our previous model Pato1 ; Pato2 by writing

[TABLE]

and

[TABLE]

where $Q_{i}=1-P_{i}.$ It is easily verified that $H=H^{GOE}$ for $\lambda=1.$

The joint probability distribution of matrix elements can be put in the form Pato1 ; Alberto

[TABLE]

with the parameter $\lambda$ being given in terms of $\alpha$ and $\beta$ by

[TABLE]

Statistical measures of the completely uncoupled $m$ blocks have been derived. They show that level repulsion disappears which can be understood since eigenvalues from different blocks behave independently. In fact, as $m$ increases the Poisson statistics are gradually approached. In the interpolating situation of partial coupling, some approximate analytical results have been derived. In Ref. Alberto , for instance, it has been found that the density $\rho(E)$ for arbitrary $\lambda$ and $m$ is given by

[TABLE]

where

[TABLE]

is Wigner’s semi-circle law with $a=\sqrt{N/\alpha}$ and

[TABLE]

Eq. (5) can be used to calculate exactly analytically the NND for $2\times 2$ and $3\times 3$ matrices Carneiro:1991 . For the $2\times 2$ case the DGOE, Eq. (5), gives

[TABLE]

where $I_{0}$ is the modified Bessel function, whose asymptotic form is

[TABLE]

Thus,there is no level repulsion for $\beta\rightarrow\infty$, $P_{2\times 2}(s,\infty)=\frac{2}{\pi}\alpha\exp\left(-\frac{\alpha}{2}s^{2}\right)$, which represents the 2x2 Poisson distribution where the usual exponential is replaced by a Gaussian. The prefactor is just 1 if 2$\alpha$ is taken to be $\pi$. In the opposite limit, $\beta\rightarrow 0$, $I_{0}(x)\approx 1-x^{2}/4$ and one obtains the Wigner distribution,

[TABLE]

Note that the parameter $\lambda$ of eq (6) is 0 if $\beta$ is $\infty$ and 1 if $\beta$ is 0.

For higher dimensions Eq. (5) can only be used for numerical simulations. This is what we are now reporting, using 2 and 3 bolck matrices of sizes 105 x 105 and 70 x 70 each, respectively. The size of the whole matrix is 210 x 210. Further, we take an ensemble of 1000 elements and fix $\alpha$ to be 1. We apply our model to analyse the eigenfrequency data of the elastomechanical vibrations of an anisotropic quartz block used in Elleg .

In this reference in order to break the flip symmetry of the crystal block gradually they removed an octant of a sphere of varying size at one of the corners. The rectangular quartz block has the dimensions $14\times 25\times 40\,mm^{3}$. The radii of the spheres containing the octants are $r=0.0,0.5,0.8,1.1,1.4$ and $1.7\,mm$ representing figures $(a)-(f)$. Figs. $1x$ and $2x$ of Ref. Elleg correspond to an octant of a huge sphere of radius $r=10.0\,mm$, whose center is inside the crystal and close to one of the corners. They found 1424, 1414, 1424, 1414, 1424 and 1419 frequency eigenmodes in cases $(a)-(f)$, respectively. These eigenfrequecies were measured in the frequency range between 600 and 900 kHz. Thus the average spacing between the modes is about 214Hz. The histograms and circles in the two figures of Ref. Elleg represent the short-range nearest-neighbor distributions $P(s)$ (Fig. 1) and the long range $\Delta_{3}(L)$ statistics (Fig. 2). In our DGOE simulation the unfolding of the calculated spectra is performed with the DGOE density given by Eq. (7) above.

In figures 1 and 2, we show the results of our simlulations as compared to the data of Ellegaard et al. Elleg for the spacing distribution and in figures 3 and 4 the long range correlation exemplified by the spectral rigidity $\Delta_{3}(L)$. We simulate the gradual breaking of the 2- or 3-fold symmetry by changing the value of the parameter $\lambda$ above. We see clearly that in so far as the $\Delta_{3}(L)$ is concerned a 3-GOE description works much better than a 2-GOE one. It is clear, however that both descriptions fall below the data, specially at large $L$. We shall analyse this discrepancy in the following using the missing level effectML .

It is often the case that there are some missing levels in the statistical sample analysed. Such a situation was addressed recently by Bohigas and Pato ML who have shown that if $g$ fraction of the levels or eigenfrequencies is missing, the $\Delta_{3}(L)$ becomes

[TABLE]

The presence of the linear term, even if small, could explain the large $L$ behavior of the measured $\Delta_{3}(L)$. We call this effect the Missing Level (ML) effect. Another possible deviation of $\Delta_{3}$ could arise from the presence of pseudo-integrable effect (PI) Abul ; bis2 . This also modifies $\Delta_{3}$ by adding a Poisson term just like Eq. (13). In the following we show that there is no need for the PI effects to explain the large-L data on the $\Delta_{3}$ if the ML effect is taken into account.

We take a study case Figs. 3b and 4b which correspond to $r=0.5\,mm$ and where $1414$ frequency eigenvalues were found. We consider this a potential ML case and take for $\Delta_{3}$, the expression given in Eq. (13) and apply to our simulations. We find perfect fit to the data, if $g$ is taken to be $0.1$, namely only $90\%$ of the eigenfrequencies were in fact taken into account in the statistical analysis. There is, threfore, room to account much better for all cases (Fig. $2a$, $2c$, $\ldots$ ) by appropiately choosing the correponding value of $g$. We have also verified that if a 2GOE description is used, namely, $m=2$ , then an account of the large-$L$ behaviour of $\Delta_{3}$ can also be obtained if a much larger number of levels were missing in the sample. In our particular case of Fig. 2b, we obtained $g=0.3$. This is 3 times larger than the ML needed in the 3GOE description. We consider the large value of $g$ needed in the 2GOE description, much too large to conform to the reported data in Ref Elleg . Figure 5 summarizes our the above.

It is therefore clear that the 3GOE description of the spectral rigidity of the eifenfrequency spectra of Elleg for the crystal block does work very well if a small fraction of the levels is taken to be missing, without resort to pseudointegrable trajectories or levels that do not feel the symmetry breaking Abul . On the other hand, the 2GOE description, which does as good as the 3GOE one in fitting the measured $P(s)$, fails dramatically in accounting for the spectral rigidity, even if as much as 30 per cent of the levels are taken as missing.

In conclusion, a random matrix model to describe the coupling of $m$-fold symmetry is constructed. The particular threefold case is used to analyse data on eigenfrequencies of elastomechanical vibration of a anisotropic quartz block. By properly taking into account the ML effect we have shown that the quartz block could very well be described by 3 uncoupled GOE’s , which are gradually coupled by the breaking of the three-fold symmetry (through the gradual removal of octants of increasing sizes), till a 1GOE situation is attained. This, therefore, indicates that the unperturbed quartz block may possess another symmetry, besides the flip one. A preliminary version of the formal aspect of this work has previously appeared in last .